Theorem xrmaxleim 11409

Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)

Assertion

Ref	Expression
xrmaxleim	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Proof of Theorem xrmaxleim

Dummy variables f g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 9872	. . . 4 |- ( ( f e. RR* /\ g e. RR* ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		simplr 528	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. RR* )
4		prid2g 3727	. . . 4 |- ( B e. RR* -> B e. { A , B } )
5	3, 4	syl 14	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. { A , B } )
6		simpllr 534	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> A <_ B )
7		xrlenlt 8091	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
8	7	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( A <_ B <-> -. B < A ) )
9	6, 8	mpbid 147	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < A )
10		breq2 4037	. . . . . . 7 |- ( y = A -> ( B < y <-> B < A ) )
11	10	notbid 668	. . . . . 6 |- ( y = A -> ( -. B < y <-> -. B < A ) )
12	11	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( -. B < y <-> -. B < A ) )
13	9, 12	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < y )
14		xrltnr 9854	. . . . . 6 |- ( B e. RR* -> -. B < B )
15	14	ad4antlr 495	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < B )
16		breq2 4037	. . . . . . 7 |- ( y = B -> ( B < y <-> B < B ) )
17	16	notbid 668	. . . . . 6 |- ( y = B -> ( -. B < y <-> -. B < B ) )
18	17	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> ( -. B < y <-> -. B < B ) )
19	15, 18	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < y )
20		elpri 3645	. . . . 5 |- ( y e. { A , B } -> ( y = A \/ y = B ) )
21	20	adantl 277	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> ( y = A \/ y = B ) )
22	13, 19, 21	mpjaodan 799	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> -. B < y )
23	2, 3, 5, 22	supmaxti 7070	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> sup ( { A , B } , RR* , < ) = B )
24	23	ex 115	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   {cpr 3623   class class class wbr 4033   supcsup 7048   RR*cxr 8060   < clt 8061   <_ cle 8062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991  ax-pre-apti 7994

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-iota 5219  df-riota 5877  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067

This theorem is referenced by:  xrmaxltsup  11423  xrmaxadd  11426  xrmineqinf  11434